2014/11/02 by Oleksandr Misiats, Misiats, Oleksandr, Oleksandr Stanzhytsyi +3 · 1 citation
Computer Science · Economics, Econometrics and Finance · Engineering · Mathematics · #35 #60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1411.0298
openalex publication_date 2014/11/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In this paper we investigate the long-time behavior of stochastic\nreaction-diffusion equations of the type du = (Au + f(u))dt + \σ(u)\ndW(t), where A is an elliptic operator, f and \σ are nonlinear maps\nand W is an infinite dimensional nuclear Wiener process. The emphasis is on\nunbounded domains. Under the assumption that the nonlinear function f\npossesses certain dissipative properties, this equation is known to have a\nsolution with an expectation value which is uniformly bounded in time. Together\nwith some compactness property, the existence of such a solution implies the\nexistence of an invariant measure which is an important step in establishing\nthe ergodic behavior of the underlying physical system. In this paper we expand\nthe existing classes of nonlinear functions f and \σ and elliptic\noperators A for which the invariant measure exists, in particular, in\nunbounded domains. We also show the uniqueness of the invariant measure for an\nequation defined on the upper half space if A is the Shr "odinger-type\noperator A = \(1)/(\ρ)(÷ \ρ \∇ u) where \ρ =≠-|x|2 is the Gaussian weight.\n